Given the following axioms:
Axiom 1: Each game is played by two distinct teams
Axiom 2: There are at least four teams
Axiom 3: There are at least six games played
Axiom 4: Each team played at most 4 games.
I need to come up with 1 to 2 more theorems. The first theorem I know is true, the second I'm unsure about and I need help developing a third.
Theorem 1: If there are exactly 4 teams, then there are at most 8 games.
Theorem 2: If there are exactly 6 games, then there are at most 12 teams.
Theorem 3:
Theorem 3: If there are exactly 3 teams, then there are at most 6 games.
To prove this theorem, we can use the given axioms and logical reasoning:
Given Axiom 1, we know that each game is played by two distinct teams. Therefore, for each game, there must be two teams involved.
To maximize the number of games, each team should play against every other team exactly once. Let's assume we have three teams: Team A, Team B, and Team C.
Based on Axiom 1, we can create the following games:
1. Team A vs. Team B
2. Team A vs. Team C
3. Team B vs. Team C
So far, we have three games.
To maximize the number of games, we can't have any more games, as each team can play at most 4 games according to Axiom 4. Adding another game involving any of the teams would result in at least one team playing more than 4 games, which violates Axiom 4.
Therefore, with three teams, we have three games, and we have proved Theorem 3: If there are exactly 3 teams, then there are at most 6 games.